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Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties. (English) Zbl 1215.14046

In this paper, higher dimensional Enriques varieties are introduced and studied. An Enriques variety is a connected compact smooth complex Kähler manifold \(Y\), such that the canonical bundle \(K_Y\) has finite order \(d\) in \(Pic(Y)\), the holomorphic Euler characteristic \(\chi(Y,\mathcal{O}_Y)\) equals one and the fundamental group \(\pi_1(Y)\) is cyclic of order \(d\). The integer \(d\) is called the index of \(Y\). An Enriques variety is called irreducible if the holonomy representation of its universal cover is irreducible. The notion generalizes the classically known Enriques surfaces.
It turns out that every irreducible Enriques variety of index \(2\) is a quotient of a Calabi-Yau manifold by a fixed point free involution, and every irreducible Enriques variety of index at least 3 is the quotient of an irreducible symplectic manifold by an automorphism acting freely.
Examples of Enriques varieties of index 3 and 4 are given using quotients of generalized Kummer varieties.

MSC:

14J50 Automorphisms of surfaces and higher-dimensional varieties
14C05 Parametrization (Chow and Hilbert schemes)
14J28 \(K3\) surfaces and Enriques surfaces
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